Geodesic
Dimension Theory Via Bisector Chains
Canadian mathematical bulletin, Tome 20 (1977) no. 3, pp. 313-317

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For two subsets Z and Y of a metric space (X, d) the set Z is said to be a bisector in Y iff Z ⊂ Y and there exist two distinct points y 1, y 2 ∊ Y such that Z = {z: d(z, y 1) = d(z, y 2) and z ∊ Y}. Considering chains of consecutive bisectors X ⊃ X 1 ⊃ ... ⊃ Xk we denote by b(X, d) the maximum of their length. The topological invariant b(X) is defined as the minimum of b(X, d) taken over the set of all metrizations of X. It is proved that if X is compact then dim(X) ≤ b(X) ≤ 2 dim(X) + 1, b(X) = 0 iff dim(X) = 0 and b(X) = n implies dim(X) = n for n = 1 and ∞. The sharp result b(En ) = n for n = 1, 2, ... is obtained for Euclidean space En . 
Janos, Ludvik. Dimension Theory Via Bisector Chains. Canadian mathematical bulletin, Tome 20 (1977) no. 3, pp. 313-317. doi: 10.4153/CMB-1977-048-3
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     author = {Janos, Ludvik},
     title = {Dimension {Theory} {Via} {Bisector} {Chains}},
     journal = {Canadian mathematical bulletin},
     pages = {313--317},
     year = {1977},
     volume = {20},
     number = {3},
     doi = {10.4153/CMB-1977-048-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1977-048-3/}
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